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As we package an entire axiom system into a schemata// we explain the difference between Successor and Math Induction, like never before
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Archimedes Plutonium
2017-08-12 18:25:31 UTC
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Newsgroups: sci.math
Date: Fri, 11 Aug 2017 06:13:01 -0700 (PDT)

Subject: Packaging all the axioms of arithmetic and algebra into a matrix columns
From: Archimedes Plutonium <***@gmail.com>
Injection-Date: Fri, 11 Aug 2017 13:13:01 +0000


Packaging all the axioms of arithmetic and algebra into a matrix columns

In these columns of 2byN matrices, I propose that they alone are all the axioms that embody Arithmetic and Algebra. Here forth, we dispense of the Peano axioms for Counting Numbers and simply cite these Matrices as the axioms of both Arithmetic and Algebra. Now, I propose for Geometry, the axioms are all embodied by a rectangle with its two diagonals and the triangles of right triangles with Pythagorean Theorem. What is true of those figures is all the axioms of Geometry, two points determine a line, opposite sides parallel due to 90 degree perpendicular. Now, the real challenging Schemata for all the axioms of a subject, is logic, where, if we can do a Arithmetic schemata of all the axioms of Arithmetic, we should be able to do a schemata of all the Logic axioms, the if then, or, and, equal, etc.

So, these schemata replace the axioms of Arithmetic and Algebra, replace the axioms of Geometry by a rectangle schemata, and replace the axioms of Logic, by a truth table schemata of the 4 possibilities::

Logic Schemata of all its axioms
TTTT, TFFF, TFFT, TFUU

Arithmetic and Algebra schemata of all its axioms


0  1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0   11
1   10
2    9
3    8
4    7
5    6


0    12
1    11
2    10
3      9
4      8
5      7
6      6

Now, in another post, I show how the Matrix Column Schemata delivers Mathematical Induction axiom, and easily delivers Successor Axiom. What the Old Math Peano axioms fail in providing is uniqueness of sum or product, fail in Complementarity , and fail in Symmetry.

Now, to package all the Geometry axioms, package them into a Schemata, we need just a figure of the Rectangle with its diagonals, for that provides us with the usual axioms of two points determine a line, the parallel postulate and most important the Pythagorean theorem. But it provides something special that Euclid and Hilbert axioms missed, for they do not axiomatize angles, provide for angles as axioms in their geometry, nor do their axioms provide for a Coordinate System, which a Rectangle as a Square provides for a axiom of Coordinate System.

___________
|                |
|                |
|__________|

when diagonals are included and angle notation, we thus provide all of Geometry Axioms that exist with a Schemata. So in the proof of a geometry theorem, when we come to a justification such as parallel line axiom, we can cite that, or we can cite rectangle-schemata.

Now, some will wonder how do they represent Hyperbolic geometry or Elliptic geometry with a Axiom Schemata? And I would answer them, that there never was three independent geometries where the parallel postulate is different, with greater than, less than or equal to 180 degrees triangles. The triangle on the sphere globe of 90 + 90 +90. There is no separate independent three geometries. All geometries are just plain Euclidean geometry. For, when you take a block and carve out a sphere representing Elliptic geometry and the hole in the block where the sphere is carved out of is Hyperbolic geometry, we realize that Elliptic and Hyperbolic were cut-outs, always cut-outs of Euclidean geometry.

The formula is true as ever-- Euclidean Geometry = Elliptic unioned with Hyperbolic geometry.

Old Math was mesmerized with the Lobachevskian and Bolyai NonEuclidean hyperbolic geometries of circa 1830, and Riemann with elliptic geometry circa 1854.

These are not independent separate geometries and do not deserve separate axioms, for they are complementaries of one another that are cut-outs of just plain Euclidean Geometry.

So, in our Rectangle Schemata, should we draw a diagonal curved arc, then the two triangles formed, one is hyperbolic and the other is elliptic, but both together are just Euclidean.

So, here, what I am doing is distilling a Schemata, that embodies all the axioms of a given Subject matter, be it Arithmetic, Algebra, Geometry, and even Logic. A schemata that one just refers to the schemata for the axiom that covers a line of a proof justification.

And this is beautiful way of announcing the axioms of Counting Numbers, for we thus know symmetry is embodied, that uniqueness is embodied, that Complementarity is embodied. And we can spell out specific traits like commutative or mathematical induction, or, successor.

AP

On Friday, August 11, 2017 at 4:21:06 PM UTC-5, Archimedes Plutonium wrote:

How is mathematical induction a feature of the schemata Re: Packaging all the axioms of arithmetic and algebra into a matrix columns
In these columns of 2byN matrices, I propose that they alone are all the axioms that embody
(snipped)

Alright, what I was saying is that this Schemata is containing all the All the Axioms embodied in the subject of both Arithmetic and Algebra. If you need a axiom for a proof, and if you find that feature trait in this schemata, then the schemata justifies your step of the proof.
- show quoted text -
Now, let us see how Mathematical Induction is a feature of that schemata.

AP

Newsgroups: sci.math
Date: Fri, 11 Aug 2017 22:02:08 -0700 (PDT)

Subject: getting Math Induction from the Columns Re: Packaging all the axioms
of arithmetic and algebra into a matrix columns
From: Archimedes Plutonium <***@gmail.com>
Injection-Date: Sat, 12 Aug 2017 05:02:08 +0000


getting Math Induction from the Columns Re: Packaging all the axioms of arithmetic and algebra into a matrix columns
Can we see how given N=10 it implies N=11, or given N=11 implies N=12.
So what is Mathematical Induction, really? Is it not the idea that if you have a property true for 1 and if you chose any other large number N, assume the property holds for N, and then show that the property holds for N+1, that what you have are all the Counting Numbers and all the Counting Numbers hold true for the property.

I like to think of Math Induction as another method of crafting the Counting Numbers. The first method is to start with 1 and keep adding 1 to it building 2, 3, 4, 5, etc. A second method is you have 1, now you assume another number N is a counting number, and if you show that its neighbor N+1 is a counting number, well, you have shown these numbers are Counting numbers. And there is no hole or gap, because N is "in general". N can be 2 or some huge number. N can be 3, or 4, or 5, or any number and if its neighbor N+1 is a counting number, then you have the set of all counting numbers.

So, to me, Mathematical Induction is an alternative means of building the Counting Numbers.

So do the Matrix Columns have the Mathematical Induction? Do the Columns suggest that given an N, it implies a N+1.

Well, I am certain the Columns obey the Successor function method of building the Counting Numbers, as you can see, each higher number is added a 1 from previous number. So if the Columns display a building of the Counting Numbers by adding 1, and the Successor function is a alternate method of building from Mathematical Induction, means the Columns have Mathematical Induction embedded in them.

And, maybe the Columns can explain Math Induction, far better than all words of the past has done in explaining. Here, the explaining of Math Induction is enhanced as meaning that just because you have a column for N, you can always increase that column to N+1, regardless if N is odd or even.
- hide quoted text -

Arithmetic and Algebra schemata of all its axioms


0  1

0  2
1  1


0  3
1  2

0  4
1  3
2  2


0  5
1  4
2  3


0  6
1  5
2  4
3  3

0  7
1  6
2  5
3  4


0  8
1  7
2  6
3  5
4  4

0  9
1  8
2  7
3  6
4  5


0  10
1   9
2   8
3   7
4   6
5   5


0   11
1   10
2    9
3    8
4    7
5    6


0    12
1    11
2    10
3      9
4      8
5      7
6      6

AP

Newsgroups: sci.math
Date: Sat, 12 Aug 2017 11:19:30 -0700 (PDT)

Subject: found the Mathematical Induction as multiplication Re: Packaging all
the axioms of arithmetic and algebra into a matrix columns
From: Archimedes Plutonium <***@gmail.com>
Injection-Date: Sat, 12 Aug 2017 18:19:30 +0000


found the Mathematical Induction as multiplication Re: Packaging all the axioms of arithmetic and algebra into a matrix columns
Now in the proof of Goldbach we recognize that the lined up Bertrand primes in each column such as 3+5 with 3*5 in the 8 column are complements, 8 complement to 15. So we have to ask whether the Goldbach is the only spot in mathematics where complements of a+b with a*b comes to prominence. Surely there must be other topics in math where complements serve a major role.
Alright, I made a beautiful breakthrough in mathematics of the Peano Axioms. And I could have only done it by challenging the axioms with the Matrix Columns.

If you notice, very carefully that in Old Math, there really is no difference between Successor Function and Mathematical Induction. So that one of them is rather superfluous. You can get Math Induction by just Successor, or, you can get Successor by just Math Induction, and that you do not need both as axioms. But the reason Old Math was convinced you needed both, was that, they felt you could not say -- any set that looks and feels like the Counting Numbers, could not be differentiated apart from the Counting Numbers of Successor Function, unless you had Math Induction telling you so.

Well, here I discovered something truly remarkable. That you do in fact need both Successor and Math Induction, but not for the silly reasoning of Old Math.

In these columns notice that you get both Successor and Math Induction and both are vastly different::
- show quoted text -
The Successor is seen as the adding of 1 so we move from 10 Column to 11 Column by addition, by adding one more new number. This is Addition and Successor. But what is Math Induction, if not successor?

Beautifully, Math Induction is going from 10 Column to 11 Column and then to 12 Column by Multiplication. So Successor is addition but Math Induction is multiplication, for look at the Even Columns are perfect-squares of 1^2, then 2^2, then 3^2, etc etc. Math Induction is the multiplication increase by one more.

So, a axiom system of the Counting Numbers, requires a Successor of addition, but also a Successor of Multiplication which we call Mathematical Induction.

AP
Archimedes Plutonium
2017-08-12 21:30:02 UTC
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Notice the matrix columns have a blank so you can write either add + or multiply

0 1


0 2
1 1


0 3
1 2


0 4
1 3
2 2

So i can fill the gap with either + or x

So we see Successor function Math Induction as just adding one more row, for example from 4 to 5

0 5
1 4
2 3

But we need a Math Induction Multiply and that is what Peano spoke of-- true for 1, assume true for N if it implies N+1 then true for all N.

0 6
1 5
2 4
3 3

Notice in the multiply of successive Even columns is the counting numbers squared-- so that multiplication in Columns is Math Induction, different from math induction add.

AP
b***@gmail.com
2017-08-12 22:40:15 UTC
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blank as in brain fartos brain?
Post by Archimedes Plutonium
Notice the matrix columns have a blank so you can write either add + or multiply
0 1
0 2
1 1
0 3
1 2
0 4
1 3
2 2
So i can fill the gap with either + or x
So we see Successor function Math Induction as just adding one more row, for example from 4 to 5
0 5
1 4
2 3
But we need a Math Induction Multiply and that is what Peano spoke of-- true for 1, assume true for N if it implies N+1 then true for all N.
0 6
1 5
2 4
3 3
Notice in the multiply of successive Even columns is the counting numbers squared-- so that multiplication in Columns is Math Induction, different from math induction add.
AP
Archimedes Plutonium
2017-08-13 17:57:40 UTC
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3+5 related to 3*5 is calculus Re: where else does complementarity pop up in math

- show quoted text -
Alright, here I am asking if math is soaked through, deep soaking of the idea of addition complementary to multiplication. Does that arise just in rare cases? Or is math full of that complementary. For it seems to us, observers, that 15 is not connected to 8 in any sort of profound manner. That 8 is no more important to 15, than is 79 or 222. So what gives here. Is math saturated full of where a+b is related intricately with a*b or is it just on rare occasions like Goldbach proof.

Okay, well, I found the answer. And it is vastly surprising. That 8 is especially connected to 15, or another example 10 is especially connected to 21 in 3+7 is to 3*7. This Complementarity is saturated full in Calculus. In Calculus, we are dealing with, primarily dy and dx, and dealing with dy +dx which is 1/2 a rectangle perimeter whilst the dy*dx is the well known integral of Calculus.

It is that we seldom if ever focus and reflect upon dy+dx, and no-one until now recognizes or sees the dy+dx as the 1/2 perimeter of the rectangle used to produce the integral of calculus.

So when I asked does 3+5 =8 and 3*5=15, what is the relationship of 8 to 15 have any math significance, well, all of calculus is thoroughly dependent on that Complementarity of 8 versus 15. For the 3 is one side of the rectangle, the 5, the other side for a 1/2 perimeter and the Calculus integral is the area 3*5= 15.

In my last edition of Correcting Math, I reached a chapter on dy+dx, but I never expounded on that concept, as well, and as in depth, as I have just now expounded.

AP

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